The Logarithm, Exponential, and Improper Integrals

The fundamental theorem guarantees that every continuous function has an antiderivative, even when no elementary formula exists. This makes the integral a tool for defining functions. The logarithm and exponential, taken for granted in calculus, are built here from the integral of $1/t$. Improper integrals then push integration past its original setting — bounded functions on bounded intervals — to unbounded domains and unbounded integrands.

The logarithm as an integral

Define a function by the area under the hyperbola $1/t$, starting at $t=1$.

Its defining properties come straight from the integral.¹

• Base point and derivative. $\ln 1=0$, and by the differentiation form $\ln$ is differentiable with $\frac{\mathbf{d}}{\mathbf{d}x}\ln x=\frac{1}{x}$.
• Monotone bijection. Since ${\ln }^{\prime }x=1/x>0$, $\ln$ is strictly increasing, hence injective, and $\ln :(0,\infty )\to \mathbb{R}$ is onto with ${\lim }_{x\to 0^{+}}\ln x=-\infty$ and ${\lim }_{x\to \infty }\ln x=+\infty$.
• Functional equation. $\ln (xy)=\ln x+\ln y$, proved by the substitution $u=yt$ in the defining integral.
• Powers. $\ln (x^q)=q\ln x$ for rational $q$.

Surjectivity uses that $\ln 2={\int }_1^2\frac{1}{t}\mathbf{d}t\ge \frac{1}{2}$, so $\ln (2^n)=n\ln 2\to \infty$; the Archimedean property and the intermediate value theorem then hit every real value.¹ Uniqueness is a corollary of the evaluation form: any function with $L(1)=0$ and $L^{\prime }=1/x$ must equal ${\int }_1^x\frac{1}{t}\mathbf{d}t$.

The exponential as the inverse

Because $\ln$ is a strictly monotone differentiable bijection, it has a differentiable inverse.

The inverse-function rule turns ${\ln }^{\prime }x=1/x$ into the defining differential equation of the exponential.¹

The two functions are mirror images across the diagonal, with reciprocal slopes at corresponding points — a slope $1/x$ for $\ln$ at $x$ becomes a slope $x$ for $\exp$ at $\ln x$.

With $\exp$ in hand, irrational powers can be defined: for $x>0$ set $x^y:=\exp (y\ln x)$, extending the rational-exponent rules by continuity, and $e^x=\exp (x)$ since $\ln e=1$.¹ The general power and exponential laws follow by differentiating the composition:

the first fixing $y$ and varying $x$, the second fixing the base $b$. The classical limit definition of the exponential is recovered from the logarithm: since $\ln$ has derivative $1$ at $x=1$,

the continuous-compounding limit, obtained by taking logarithms and using $n\ln (1+\frac{x}{n})\to x$.¹

Improper integrals

The Riemann integral is defined only for bounded functions on bounded intervals. Two extensions, each a limit of proper integrals, cover unbounded domains and unbounded integrands.

The prototype family sets the convergence threshold. The exponent $p$ determines whether the tail area is finite.²

At infinity the integrand must decay faster than $1/x$; near $0$ the singularity must be milder than $1/x$.

The comparison test

Convergence of a hard integral often follows from a simpler dominating one, just as for series.

Splitting an improper integral before establishing convergence is invalid: ${\int }_2^{\infty }\frac{2}{x^2-1}\mathbf{d}x$ converges, but rewriting it as $\int \frac{1}{x-1}-\int \frac{1}{x+1}$ produces $\infty -\infty$, since each piece diverges.²

Absolute versus conditional convergence

An improper integral can converge through cancellation without converging absolutely — the continuous analogue of a conditionally convergent series.

The alternating signs of $\sin x$ provide the same cancellation as in the alternating harmonic series.

The integral test for series

A decreasing nonnegative function ties the convergence of a series to the convergence of an integral, because its terms and its area sandwich each other.

The estimate is quantitative, not just qualitative: it brackets a sum to within $f(k)$.

The same test recovers the $p$-series result — $\sum n^{-p}$ converges exactly when ${\int }_1^{\infty }{x}^{-p}\mathbf{d}x$ does, i.e. for $p>1$.